VI-DGP: A variational inference method with deep generative prior for solving high-dimensional inverse problems

TL;DR

This paper introduces VI-DGP, a novel variational inference approach utilizing deep generative priors and physics-informed surrogates to efficiently solve high-dimensional Bayesian inverse problems involving PDEs.

Contribution

It proposes a deep generative prior-based variational inference method with physics-informed surrogates for improved accuracy and efficiency in high-dimensional inverse problems.

Findings

Demonstrates high accuracy in log-permeability estimation

Achieves significant computational efficiency improvements

Validates method on flow in heterogeneous media

Abstract

Solving high-dimensional Bayesian inverse problems (BIPs) with the variational inference (VI) method is promising but still challenging. The main difficulties arise from two aspects. First, VI methods approximate the posterior distribution using a simple and analytic variational distribution, which makes it difficult to estimate complex spatially-varying parameters in practice. Second, VI methods typically rely on gradient-based optimization, which can be computationally expensive or intractable when applied to BIPs involving partial differential equations (PDEs). To address these challenges, we propose a novel approximation method for estimating the high-dimensional posterior distribution. This approach leverages a deep generative model to learn a prior model capable of generating spatially-varying parameters. This enables posterior approximation over the latent variable instead of the complex parameters, thus improving estimation accuracy. Moreover, to accelerate gradient computation, we employ a differentiable physics-constrained surrogate model to replace the adjoint method. The proposed method can be fully implemented in an automatic differentiation manner. Numerical examples demonstrate two types of log-permeability estimation for flow in heterogeneous media. The results show the validity, accuracy, and high efficiency of the proposed method.